Numerical renormalization group method for quantum impurity systems

Bulla, Ralf; Costi, T. A.; Pruschke, Thomas

doi:10.1103/revmodphys.80.395

Cited by 1,454 publications

(2,094 citation statements)

References 321 publications

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“…To solve the impurity problem, various numerical methods have been proposed: iterated perturbation theory, continuous time quantum Monte Carlo, exact diagonalization (ED), 37) non-crossing approximation (NCA), [38][39][40][41] numerical renormalization group (NRG), 42) etc. In the following section, we briefly describe IPT and CT-QMC, which we used in this study.…”

Section: Impurity Solversmentioning

confidence: 99%

DMFT Study on the Electron–hole Asymmetry of the Electron Correlation Strength in the High T_c Cuprates

2017

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Recent experiments revealed a striking asymmetry in the phase diagram of the high temperature cuprate superconductors. The correlation effect seems strong in the hole-doped systems and weak in the electrondoped systems. On the other hand, a recent theoretical study shows that the interaction strengths (the Hubbard U ) are comparable in these systems. Therefore, it is difficult to explain this asymmetry by their interaction strengths. Given this background, we analyze the one-particle spectrum of a single band model of a cuprate superconductor near the Fermi level using the dynamical mean field theory. We find the difference in the "visibility" of the strong correlation effect between the hole-and electron-doped systems. This can explain the electron-hole asymmetry of the correlation strength without introducing the difference in the interaction strength.

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Section: Impurity Solversmentioning

confidence: 99%

DMFT Study on the Electron–hole Asymmetry of the Electron Correlation Strength in the High T_c Cuprates

2017

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“…[14] for a review. In parallel, many theoretical works have followed, and at present reliable methods, such as the Numerical Renormalization Group (NRG) [15,16], have been developped. These calculations can allow quantitative comparison to experiments, especially because the spin S = 1/2 Kondo effect shows simple universal scaling laws, that can be checked in quantum transport measurements.…”

Section: Introductionmentioning

confidence: 99%

Universal transport signatures in two-electron molecular quantum dots: gate-tunable Hund’s rule, underscreened Kondo effect and quantum phase transitions

Florens

Freyn

Roch

et al. 2011

J. Phys.: Condens. Matter

110

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Abstract. We review here some universal aspects of the physics of two-electron molecular transistors in the absence of strong spin-orbit effects. Several recent quantum dots experiments have shown that an electrostatic backgate could be used to control the energy dispersion of magnetic levels. We discuss how the generically asymmetric coupling of the metallic contacts to two different molecular orbitals can indeed lead to a gate-tunable Hund's rule in the presence of singlet and triplet states in the quantum dot. For gate voltages such that the singlet constitutes the (non-magnetic) ground state, one generally observes a suppression of low voltage transport, which can yet be restored in the form of enhanced cotunneling features at finite bias. More interestingly, when the gate voltage is controlled to obtain the triplet configuration, spin S = 1 Kondo anomalies appear at zero-bias, with non-Fermi liquid features related to the underscreening of a spin larger than 1/2. Finally, the small bare singlet-triplet splitting in our device allows to fine-tune with the gate between these two magnetic configurations, leading to an unscreening quantum phase transition. This transition occurs between the non-magnetic singlet phase, where a two-stage Kondo effect occurs, and the triplet phase, where the partially compensated (underscreened) moment is akin to a magnetically "ordered" state. These observations are put theoretically into a consistent global picture by using new Numerical Renormalization Group simulations, taylored to capture sharp finie-voltage cotunneling features within the Coulomb diamonds, together with complementary outof-equilibrium diagrammatic calculations on the two-orbital Anderson model. This work should shed further light on the complicated puzzle still raised by multi-orbital extensions of the classic Kondo problem.

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“…The numerical calculation is carried out by Wilson's numerical renormalization group (NRG) method using a recurrence relation at each renormalization step N . [32][33][34] The original Hamiltonian is related to the N th NRG Hamiltonian:…”

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SU(2)–SU(4) Kondo Crossover and Emergent Electric Polarization in a Triangular Triple Quantum Dot

2016

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We study an orbitally degenerate Kondo effect in a triangular triple quantum dot (TTQD), where the three dots are connected vertically with a single metallic lead through electron tunneling. Both spin and orbital degrees of freedom play an important role in the SU(4) Kondo effect. This is demonstrated by an equilateral TTQD Kondo system at half-filling, by Wilson's numerical renormalization group method. We show how an emergent electric polarization of the TTQD is associated with a crossover from SU(4) to SU(2) symmetry in the low-temperature state. A marked sign reversal of the electric polarization is generated by the fine-tuning of Kondo coupling with degenerate orbitals, which can be utilized to reveal orbital dynamics in the SU(4) Kondo effect.Since Kondo's pioneering work, 1 quantum impurities coupled to conduction electrons have been one of the major issues of strongly correlated electron systems for over half a century. 2-4 Nowadays, the accumulation of knowledge about the Kondo effect in bulk systems is applied to nanoscale or mesoscopic devices such as quantum dots (QDs) and molecular junctions, which extends the frontiers of quantum phenomena. 5-10 Indeed, typical Kondo behavior is observed in the conductance between metallic leads through a single QD, whose quantized energy levels are highly tuned by the gate voltage. 11-15 Recent nanofabrication techniques have facilitated various geometric configurations of multiple QDs. [16][17][18][19] Multicoupled QDs are considered as variants of a molecule in which spin and charge degrees of freedom play a crucial role through strong electron correlation.Such innovation of the Kondo physics motivates us to study a triangular triple quantum dot (TTQD) as the simplest multiple QD system with a closed loop. 16,18 When the three equivalent QDs form an equilateral triangle, doubly degenerate molecular orbitals of TTQD play the same role as a spin. At half-filling, the TTQD ground state is fourfold-degenerate with respect to both spin and orbital degrees of freedom, namely, SU(4)-symmetric. It is expected that the orbital dynamics leads to a highly symmetric SU(4) Kondo effect as a theoretical extension of a conventional spin SU(2) Kondo effect. 2, 3 Previous theoretical studies of TTQD systems have so far paid much attention to a lateral metallic contact that brings about rich Kondo physics related to three-site spin configurations of TTQD. [20][21][22][23][24][25][26][27][28] In such a lateral geometry, it is not practical to search for the SU(4) Kondo effect since the molecular orbital is not classified as a good quantum number. On the other hand, it is more appropriate to investigate a different TTQD system in which each QD is point-contacted vertically with a single lead. In this case, the SU(4) Kondo effect is realized by the C 3 symmetry of TTQD.In a related context, the SU(4) Kondo physics has been frequently studied using carbon nanotube devices whose dynamical properties come from valley degrees of freedom as well as spins, which correspond to the clockwis...

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Numerical renormalization group method for quantum impurity systems

Cited by 1,454 publications

References 321 publications

DMFT Study on the Electron–hole Asymmetry of the Electron Correlation Strength in the High T_c Cuprates

DMFT Study on the Electron–hole Asymmetry of the Electron Correlation Strength in the High T_c Cuprates

Universal transport signatures in two-electron molecular quantum dots: gate-tunable Hund’s rule, underscreened Kondo effect and quantum phase transitions

SU(2)–SU(4) Kondo Crossover and Emergent Electric Polarization in a Triangular Triple Quantum Dot

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Numerical renormalization group method for quantum impurity systems

Cited by 1,454 publications

References 321 publications

DMFT Study on the Electron–hole Asymmetry of the Electron Correlation Strength in the High Tc Cuprates

DMFT Study on the Electron–hole Asymmetry of the Electron Correlation Strength in the High Tc Cuprates

Universal transport signatures in two-electron molecular quantum dots: gate-tunable Hund’s rule, underscreened Kondo effect and quantum phase transitions

SU(2)–SU(4) Kondo Crossover and Emergent Electric Polarization in a Triangular Triple Quantum Dot

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DMFT Study on the Electron–hole Asymmetry of the Electron Correlation Strength in the High T_c Cuprates

DMFT Study on the Electron–hole Asymmetry of the Electron Correlation Strength in the High T_c Cuprates